. - , .
, .
1. , y p - .
. , , (x3, x2 - 1) = 1, y3 p , . . , , p x3, , y3 x3, , x = y, . . x - . , , , . . x = 2, y = 2, p = 3.
: x = 2, y = 2, p = 3.
2. .
,
. 1 , 1, -1.
:
(1) (2)
: (1) (2)
:
3. : , p - .
:
.
. , , , , , ,
(1) (2) (3) (4)
, :
(1) (2) (3) (4)
:
4. : .
:
.
18. , , 18:
18 : 1, 2, 3, 6, 9, 18. 18, - :
(1) (2) (3)
(4) (5) (6)
(1) (2) (3) (4) (5) (6)
:
5.
.
: .
:
(1) (2) (3) (4)
x y - , :
:
ax + by = c, a, b, c - ,
1. (a; b) = d, x y, .
( .)
. 1232 1672.
1) (1232, 1672):
(1232, 1672) = 88.
2) 88 , , :
|
|
, . . .
2. ax + by = l (a, b) = 1, .
1. , + by = 1, (, b) = 1, 1 b.
. 15x + 37y = 1.
1) (15, 37):
(15, 37) = 1
2) 1 , , :
, . .
x0 = 5, y0 = -2.
3. + by = (, b) = d > 1 d, .
.
. 16x - 34y = 7.
(16, 34) = 2, 7 2, .
4. + by = (a, b) = d > 1 c d, 1 + b1 = c1, (a1, b1) = 1.
, .
5. + by = (, b) = 1, :
x0, y0 - + by = 1, t - .
, -, , -, .
+ by = , (, b) = 1:
1) + by = 1 1 b ( , );
2) :
x0, y0 - + by = 1, t .
t , : , ( ) . .
1. 407 - 2816 = 33.
1) , 37 - 256 = 3.
2) 37x - 256y = 1.
.
.
3) :
:
2. , 17 9, 22 4.
17 x, : 17x + 9, . , x = 6 - ; x = 60 , x = 59 .
22 y, : 22y + 4, . x.
|
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: 17x + 9 = 22y + 4, 22y - 17x = 5 - .
1) : -17x + 22y = 1.
,
2) :
.
.
, , t = 1, x = 67, y = 52 - .
, .
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.
: 774.
[1] . , , - . - , , .
, 840 120, 140; , , 840 .